Positivity-preserving and asymptotic preserving method for 2D Keller-Segal equations

Jian-Guo Liu Duke University Li Wang State University of New York at Buffalo Zhennan Zhou Duke University

Numerical Analysis and Scientific Computing mathscidoc:1702.25021

We propose a semi-discrete scheme for 2D Keller-Segel equations based on a symmetrization reformation, which is equivalent to the convex splitting method and is free of any nonlinear solver. We show that, this new scheme is unconditionally stable as long as the initial condition does not exceed certain threshold, and it asymptotically preserves the quasi-static limit in the transient regime. Furthermore, we prove that the fully discrete scheme is conservative and positivity preserving, which makes it ideal for simulations. The analogical schemes for the radial symmetric cases and the subcritical degenerate cases are also presented and analyzed. With extensive numerical tests, we verify the claimed properties of the methods and demonstrate their superiority in various challenging applications.
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  title={Positivity-preserving and asymptotic preserving method for 2D Keller-Segal equations},
  author={Jian-Guo Liu, Li Wang, and Zhennan Zhou},
Jian-Guo Liu, Li Wang, and Zhennan Zhou. Positivity-preserving and asymptotic preserving method for 2D Keller-Segal equations. 2017. In MATHEMATICS OF COMPUTATION. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170206153355955558192.
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